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# Doodling in math: Sick number games

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Pretend you're me and you're in math class. Actually... nevermind, I'm sick so I'm staying home today so pretend you are Stanislaw Ulam instead. What I am about to tell you is a true story. So you are Stan Ulam and you're at a meeting but there's this really boring presentation so of course you're doodling and, because you're Ulam and not me, you really like numbers... I mean <i>super</i> like them. So much that what you're doodling is numbers, just counting starting with one and spiralling them around. I'm not too fluent in mathematical notation so so i find things like numbers to be distracting, but you're a number theorist and if you love numbers who am I to judge? Thing is, because you know numbers so intimately, you can see beyond the confusing, squiggly lines you're drawing right into the heart of numbers. And, because you're a number theorist, and everyone knows that number theorists are enamoured with prime numbers( which is probably why they named them "prime numbers"), the primes you've doodled suddenly jump out at you like the exotic indivisible beasts they are... So you start drawing a heart around each prime. Well... it was actually boxes but in my version of the story it's hearts because you're not afraid to express your true feelings about prime numbers. You can probably do this instantly but it's going to take me a little longer... I'm all like - "Does 27 have factors besides one and itself? ... o.0 ... Oh yeah, it's 3 times 9, not prime." "Hmmm what about 29...? pretty sure it's prime." But as a number theorist, you'll be shocked to know it takes me a moment to figure these out. But, even though you have your primes memorised up to at least 1000 that doesn't change that primes, in general, are difficult to find. I mean if I ask you to find the highest even number, you'd say, "that's silly, just give me the number you think is the highest and i'll just add 2.... BAM!!" But guess what the highest prime number we know is? 2 to the power of 43,112,609 - 1. Just to give you an idea about how big a deal primes are, the guy that found this one won a $100,000 prize for it! We even sent our largest known prime number into space because scientists think aliens will recognise it as something important and not just some arbitrary number. So they will be able to figure out our alien space message... So if you ever think you don't care about prime numbers because they're 'not useful', remember that we use prime numbers to talk to aliens, I'm not even making this up! It makes sense, because mathematics is probably one of the only things all life has in common. Anyway, the point is you started doodling because you were bored but ended up discovering some neat patterns. See how the primes tend to line up on the diagonals? Why do they do that?... also this sort of skeletal structure reminds me of bones so lets call these diagonal runs of primes: Prime Ribs! But how do you predict when a Prime Rib will end? I mean, maybe this next number is prime... (but my head is too fuzzy for now this right now so you tell me.) Anyway...Congratulations, You've discovered the Ulam Spiral! So that's a little mathematical doodling history for you. Yyou can stop being Ulam now... or you can continue. Maybe you like being Ulam. (thats fine) However you could also be Blaise Pascal. Here's another number game you can do using Pascal's triangle.(I don't know why I'm so into numbers today but I have a cold so if you'll just indulge my sick predelections maybe I'll manage to infect you with my enthusiasm :D Pascal's Triangle is the one where you get the next row in the triangle by adding two adjacent numbers. Constructing Pascal's Triangle is, in itself a sort of number game because it's not just about adding, but about trying to find patterns and relationships in the numbers so you don't have to do all the adding. I don't know if this was discovered through doodling but it was discovered independantly in: France, Italy, Persia, China and probably other places too so it's possible someone did. Right... so I don't actually care about the individual numbers right now. So, if you still Ulam, you pick a property and highlight it(e.g. if it's even or odd) If you circle all the odd numbers you'll get a form which might be starting to look familiar. And it makes sense you'd get Sierpinski's Triangle because when you add an odd number and an even number, you get an odd number. (odd + odd) = even and (even + even) = even... So it's just like the crash and burn binary tree game. The best part about it is that, if you know these properties, you can forget about the details of the numbers You don't have to know that a space contains a 9 to know that it's going to be odd. Now, instead of two colours, let's try three. we'll colour them depending on what the remainder is when you divide them by three(instead of by two). Here's a chart! :) So, all the multiples of three are coloured red, remainder of one will be coloured black and remainder of two will be coloured green. The structure is a little different from Sierpinski's Triangle already but I'm tired of figuring out remainders based of individual numbers, so Let's figure out the rules... If you add up two multiples of three you always get another multiple of three( which is the sort of fact you use everday in math class) However, here this means (red + red) = red. and when you add a multiple of three to something else, it doesn't change it's remainder. So, (red + green) = green and (red + black) = black. (remainder 1 + remainder 1) = remainder 2, (remainder 2+ remainder 2) = remainder 4 and the remainder of 4 divided by 3 is one and (1+2) = 3 remainder 0. (whew...) The bottom line is you're making up some rules as to what coloured dots combine to produce which other coloured dots and then you're following those rules to their mathematical and artistic conclusion... The numbers themselves were never necessary to get this picture. Anyway, those are just a couple of examples of number games that are out there but you should also try making up your own. For example, I have no idea what you'd get if you highlighted the prime numbers in Pascal's Triangle, maybe nothing interesting(who knows...) Or, what happens if, instead of adding to get the next row, you start with a two(and a sea of invisible ones) and multiple two adjacent numbers to get the next row. I've no idea what hapens there either or if it's already a 'thing' people do. (Hmmm? o.0 Powers of two...) I know another way to write this. Ok, that makes sense. Then there is also a thing called Floyd's Triangle where you put the numbers like this... Maybe you can do something with that as well. ... Man, it seems like everyone has a triangle these days... I'm going to take a nap... ZZZzzz...